3.3 Outlook and future work
3.3.4 Other work in different research lines
Another line of research which was developed during the PhD, other than the study of open dissipative dynamics, is the study of whether properties of the thermodynamic limit of a sequence of Hamiltonians can be inferred from the study of an increasing sequence of finite systems. More specifically, we were interested in the possible pitfalls of the common approach taken to study the large-system limit of Hamiltonian models, which consists in analyzing a sequence of finite cases, either experimentally or numerically, to then extrapolate some properties of the limit. In a number of important cases [6, 45, 52, 68, 79] this approach has been successful and has led to important insight on the properties of the physical models in the large-system limit. On the other hand, there is a general negative result: the problem of deciding whether a sequence of translation invariant local Hamiltonians is gapped or gapless in the limit is an undecidable problem [15]. This result shows that unpredictable behavior can be shown by this type of models, which lead us to be interested in exploring the possibilities of constructing such exotic examples.
The resulting work has been presented in a (yet to be published) paper [8], in which we present two families of models that show a surprising property: for any finite region smaller than a fixed threshold, the ground state and low-excited states are classical states (product states in the computational basis); above the threshold they show instead topological prop- erties, which are characteristic of quantum models. By increasing the local dimension of the spins, the threshold can be made arbitrarily large, and already for local dimension d = 10 becomes bigger than the estimated number of particles in the universe. We denoted this phenomenon size-driven phase transition, as it can be seen as an abrupt change from a classical model to a quantum model driven by the change of the system size parameter.
The two constructions are based on different ideas, and have different thresholds scaling. They are both based on tilings models: a tiling is a covering of a region of the plane with unit squares with colored edges, such that colors on neighboring squares matches. It has been shown that the problem of deciding, given a finite set of tiles, whether they can tile the whole plane or not is undecidable [9, 73]. This result was a building block of the undecidability result of [15]. We modified their construction by using plaquette and star
interactions (instead of just having plaquettes), and with this we produced two families of models. The first one is based on the idea of constructing periodic patterns with very large periods (compared to the number of colors used in the tiling), in such a way that a specific pattern only appears once every period. By penalizing that pattern, we can induce a energy frustration for every lattice size larger than the period: this allows us to implement the transition between the classical and quantum models.
The other construction is based on the idea, already present in the results of undecidabil- ity, of encoding the history of a Turing Machine into the groundstate of the Hamiltonian. In this way, by giving an energy penalty if the machine halts, we have the same phenomenon as before of an energy frustration when the system size becomes large enough for the machine to halt. Since determining whether (and when) a Turing machine will halt is an undecidable problem, this was one of the key ingredient into showing undecidability of the spectral gap. We were able to highly optimize the cost of the encoding, in the sense of the minimal local Hilbert space dimension needed to write the history state of the Turing machine into the spin model. With this optimized encoding, we considered the so called Busy Beavers Turing machines: a machine with a very small dimension, but whose halting time is surprising large - it actually grows faster than any computable function. This gives us models which have a relatively small local dimension, but for which there is a frustration only for extremely large system sizes. Again, this frustration allows us to implement the phase transition.
CAPÍTULO
4
Introducción
Esta tesis está organizada como sigue. En el apartado 4.1 definimos los principales objetos de interés, que son los semigrupos dinámicos de canales cuánticos. Recordaremos las propiedades de sus generadores, llamados Lindbladianos. Se explicará por qué son un buen modelo para evoluciones cuánticas con ruido, y se definirá el tiempo de equilibración, una propiedad central en las hipótesis necesarias para probar los resultados principales. Se presentará la relación entre tiempo de equilibración y otras importantes propiedades del semigrupo, como el gap espectral, la desigualdad de log-Sobolev e hipercontractividad. Estas conexiones nos permitirán encontrar técnicas para probar la condición de equilibración rápida. Introduciremos también la noción de información mutua y se discutirá la ley de área para estados, así como sus conexiones con la dificultad de simulación y los estados de redes tensoriales. Finalmente, se discutirá por qué la estabilidad es una condición fundamental para cualquier modelo matemático de un sistema físico. En el apartado 4.2 se definirán las hipótesis principales y una síntesis de los resultados obtenidos, junto con una breve presentación de las herramientas técnicas desarrolladas para probarlos. En el apartado 4.3 se discutirán las lineas futuras de investigación en las cuales estamos trabajando actualmente.El resto de la tesis está compuesta por las publicaciones que recogen los resultados obtenidos a lo largo del Doctorado. Los capítulos corresponden a las publicaciones sigu- ientes.
5. [S] T. S. Cubitt, A. Lucia, S. Michalakis y D. Perez-Garcia. “Stability of Local Quantum Dissipative Systems”. En: Commun. Math. Phys. 337.3 (abr. de 2015), págs. 1275-1315. DOI: 10.1007/s00220-015-2355-3
6. [R] A. Lucia, T. S. Cubitt, S. Michalakis y D. Perez-Garcia. “Rapid mixing and stability of quantum dissipative systems”. En: Phys. Rev. A (Rapid Comm.) 91.4 (abr. de 2015). DOI: 10.1103/physreva.91.040302
7. [A] F. G. S. L. Brandão, T. S. Cubitt, A. Lucia, S. Michalakis y D. Perez-Garcia. “Area law for fixed points of rapidly mixing dissipative quantum systems”. En: J. Math. Phys. 56.10 (oct. de 2015), pág. 102202.DOI: 10.1063/1.4932612
El impacto de estas publicaciones está reflejado en el número de citas recibidas, a pesar de su reciente publicación: en particular, [S] en el momento de la publicación de esta tesis ha recibido ya 18 citas, mientras que [R] ha recibido 3 y [A] una. Además, los resultados obtenidos han sido presentados como ponencia oral en las conferencias más prestigiosas del área: en Quantum Information Processing and Communications 2013 (QIPC2013), en el 17th Conference on Quantum Information Processing (QIP2014), y en Theory of Quantum Computation, Communication and Cryptography (TQC2015).